Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{t^3 - 7t^2 + 6t}{-5t^2 - 20t + 25}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {t(t^2 - 7t + 6)} {-5(t^2 + 4t - 5)} $ $ r = -\dfrac{t}{5} \cdot \dfrac{t^2 - 7t + 6}{t^2 + 4t - 5} $ Next factor the numerator and denominator. $ r = - \dfrac{t}{5} \cdot \dfrac{(t - 1)(t - 6)}{(t - 1)(t + 5)}$ Assuming $t \neq 1$ , we can cancel the $t - 1$ $ r = - \dfrac{t}{5} \cdot \dfrac{t - 6}{t + 5}$ Therefore: $ r = \dfrac{ -t(t - 6)}{ 5(t + 5)}$, $t \neq 1$